论文标题
王 - 卡萨蒂售出的柜台在可集成的台球中的增长
Growth of the Wang-Casati-Prosen counter in an integrable billiard
论文作者
论文摘要
这项工作是由Wang，Casati和Prosen [Phys的一篇文章的动机。 Rev. E Vol。 89，042918（2014）]专门研究二维非理性右三角台球中的磨性研究。那里提出的数值结果表明，这些台球通常不是奇异的。但是，当台球角度等于$π/2 $ $倍的liouvillian非理性，一种liouvillian非理性的$π/2 $倍时，它们就会变成千古来。 特别是Wang等。研究一个特殊的整数计数器，反映了对速度取向的非理性贡献；他们猜想该计数器位于通用案例中，但在liouvillian案例中生长。我们提出了Wang-Casati售出的计数器的概括：此概括允许考虑理性台球。我们表明，如果$ 45^{\ circ} \！\！ ：\！ 45^{\ circ} \！\！ ：\！ 90^{\ Circ} $台球，计数器无限期地增长，与Wang等人建议的Liouvillian场景一致。
This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that these billiards are generally not ergodic. However, they become ergodic when the billiard angle is equal to $π/2$ times a Liouvillian irrational, a Liouvillian irrational, a class of irrational numbers which are well approximated by rationals. In particular, Wang et al. study a special integer counter that reflects the irrational contribution to the velocity orientation; they conjecture that this counter is localized in the generic case, but grows in the Liouvillian case. We propose a generalization of the Wang-Casati-Prosen counter: this generalization allows to include rational billiards into consideration. We show that in the case of a $45^{\circ} \!\! : \! 45^{\circ} \!\! : \! 90^{\circ}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.
